Drawing Lines The Bresenham Algorithm for drawing lines and filling polygons

Plotting a line-segment Bresenham published algorithm in 1965 It was originally to be used with a plotter It adapts well to raster “scan conversion” It uses only integer arithmetic operations It is an “iterative” algorithm: each step is based on results from the previous step The sign of an “error term” governs the choice among two alternative actions

Scan conversion The actual line is comprised of points drawn from a continuum, but it must be “approximated” using pixels from a discrete grid.

The various cases Horizontal or vertical lines are easy cases Lines that have slope 1 or -1 are easy, too Symmetries leave us one remaining case: 0 < slope < 1 As x-coodinate is incremented, there are just two possibilities for the y-coordinate: (1) y-coordinate is increased by one; or(2) y-coordinate remains unchanged

0 < slope < 1 y increases by 1y does not change X-axis Y-axis

Integer endpoints ΔXΔX ΔYΔY slope = ΔY/ΔX (X0,Y0) (X1,Y1) ΔY = Y1 – Y0 ΔX = X1 – X0 0 < ΔY < ΔX

Which point is closer? A B x i -1 xixi y = mx + b error(A) = (y i ) – y* error(B) = y* - (y i -1 ) ideal line y i -1 y i -1 +1

The Decision Variable Choose B if and only if error(B)<error(A) Or equivalently: error(B) – error(A) < 0 Formula: error(B) – error(A) = 2m(x i – x 0 ) + 2(y 0 – y i -1 ) -1 Remember: m = Δy/Δx (slope of line) Multiply through by Δx (to avoid fractions) Let d i = Δx( error(B) – error(A) ) Rule is: choose B if and only if d i < 0

Computing d i+1 from d i d i+1 = 2(Δy)(x i+1 – x 0 ) +2(Δx)(y 0 – y i ) – Δx d i = 2(Δy)(x i – x 0 ) + 2(Δx)(y 0 – y i-1 ) – Δx The difference can be expressed as: d i+1 = d i + 2(Δy)(x i+1 – x i ) – 2(Δy)(y i – y i-1 ) Recognize that x i+1 – x i = 1 at every step And also: y i – y i-1 will be either 0 or 1 (depending on the sign of the previous d)

How does algorithm start? At the outset we start from point (x 0,y 0 ) Thus, at step i =1, our formula for d i is: d 1 = 2(Δy) - Δx And, at each step thereafter: if ( d i < 0 ) { d i+1 = d i + 2(Δy); y i+1 = y i ; } else{ d i+1 = d i + 2(Δy-Δx); y i+1 = y i + 1; } x i+1 = x i + 1;

‘bresdemo.cpp’ The example-program is on class website: It draws line-segments with various slopes The Michener algorithm (for a circle-fill) is also included, for comparative purposes Extreme slopes (close to zero or infinity) are not displayed in this demo program They can be added by you as an exercise

Filling a triangle or polygon The Bresenham’s method can be adapted But an efficient data-structure is needed All the sides need to be handled together We let the y-coordinate steadily increment For sides which are “nearly horizontal” the x-coordinates can change by more than 1

Triangle Illustration

Non-Convex Polygons

Bucket-Sort Y XLOXHI 13 17

Handling Corners

Legacy Software We have a polygon-fill demo: ‘polyfill.cpp’ You can find it now on our class website But it was written in 1997 for MS-DOS We’d like to run it on our Linux system But it will need a number of modifications This is our first programming assignment: to “adapt” this obsolete application so it can execute on a contemporary platform

What are some issues? GNU compiler enforces stricter standards Sizes of some data-types are now larger Physical VRAM now must be “mapped” I/O instructions now require permissions Real-Mode addresses must be converted Not all the header-files are still supported Interface to CPU registers is a bit different

Converting addresses TURBO-C++ allowed use of ‘far’ pointers Use a macro to create a pointer to VRAM: uchar *vram = MK_FP( 0xA000, 0x0000 ); Real-mode address has two components: (16-bit segment and 16-bit offset) For Linux we convert to a 32-bit address: uchar *vram = (uchar*)0x000A0000; Formula: address = 16*segment + offset

Hardware Issues? Pentium CPUs are “backward compatible” BIOS firmware can use Virtual-8086 mode SVGA still supports older graphics modes The keyboard’s mechanism is unchanged Feasibility test: if we will “boot” MS-DOS, we can easily run the ‘polyfill’ application So it’s just a software problem to give our MS-DOS program a new life under Linux!